There has been considerable
interest recently in the question of when, given a smooth simple closed extreme curve
in R3 as boundary, there exists an embedding of a disk having that boundary which
is minimal or minimizing in some appropriate subclass of Lipschitz mappings of the
disk. Almgren and Simon [2] showed that an area minimizing embedding of a disk
exists in the class of all Lipschitz embeddings of disks. (They also showed
that there exists an area minimizing embedding of a disk with k handles in
the class of all such embeddings in case there exists some mapping of the
disk with k handles whose area is less than that of any mapping with k − 1
handles.) Tomi and Tromba [6] showed that there exists a minimal (not
necessarily minimizing) embedding of a disk in the class of all Lipschitz
mappings of the disk. Meeks and Yau [3] have shown that there exists an area
minimizing embedding of the disk in the class of all Lipschitz mappings of the
disk.
This paper shows that if one minimizes the integral of an essentially
oriented integrand, it is possible for an immersion of the disk to have less
integral than any embedding; such integrands arbitrarily closely approximate
area.
